The Conjugacy Analysis of Modified Part of Scroll Profiles

Transcription

1 Purdue University Purdue e-pubs International Compressor Engineering Conference School of Mechanical Engineering 1994 The Conjugacy Analysis of Modified Part of Scroll Profiles Z. Liu G. Du Z. Qi J. Gu Follow this and additional works at: Liu, Z.; Du, G.; Qi, Z.; and Gu, J., "The Conjugacy Analysis of Modified Part of Scroll Profiles" (1994). International Compressor Engineering Conference. Paper This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at Herrick/Events/orderlit.html

2 THE CONJUGACY ANALYSIS OF MODIFIED PART OF SCROLL PROFILES Liu Zhenquan, Professor of Mechanical Engineering Du Guirong,Associ'ate Professor of Mechanical Engineering Qi Zhiyong, Research Assistant BY Gu J ianfeng, Research Assistant Address:Chemical Machinery Engineering Department, Gansu University of Technology, Lanzhou Gansu, P.R. China Telephone: ABSTRACT A key and very important problem in the design of a scroll compressor is the wrap modification of the scroll profile. This problem has attracted more and more attention from researchers worldwide and several methods have been used in the past. In this paper a mathematical conjugacy relation derivation and analysis of the new wrap modification method are given, based on my previous paper presented at the 1992 International Compressor Engineering Conference at Purdu [1]. It is hoped that this analysis will make the new modification method of scroll profile more reliable. NOMENCLATURE Ror Rotational radius of crank a Radius of the base circle of involute y Modified angle INTRODUCTION The general requirement for conjugacy of scroll surfaces has been defined by several researchers [2). The first condition for conjugacy is that for any given point on a working scroll surface, there is one and only one unique point on the other surface which is its conjugate. The second condition for conjugacy is that when any arbitrary conjugate pair of points is in 479

3 contact, the centers of the two scroll elements are offset by a constant distance, the orbit radius ( Ror). And the third condition for conjugacy is that at the two conjugate points, vectors tangent to the two surfaces are parallel to each other and normal to the direction of the offset of the two scroll centers. These rules can be used to govern what types of surfaces may be appropriate for scroll machines and to design scroll profiles with special properties. In recent years, in the process of theoretical study and engineering design of the scroll compressor, the authors have developed a new wrap modification method [1]. As a supplement, two kinds of mathematical derivation and analysis of the conjugacy relationship between the two modified parts of a pair of scroll profiles are presented in this paper. It is hoped that this work will make the method more reliable. t.graphic Analysis of Conjugate Relation of The Modified Part Fig. 1 shows the modified result of the involute of a circle by the new method, and it satisfies the following equations [3J: R=r+Ror r = a Ror sin2 a 2 ctg6+26 =:tty Assuming that the figure as shown in Fig. 1 represents the fixed scroll element, starting from this position and around point 0 1, an identical scroll vane is turned through 180 then an orbiting scroll member is found and shares a common center F with the fixed scroll. If the orbiting scroll moves toward the fixed scroll in any direction, an arbitrary conjugate pair of points is in contact, the centers of the two arcs are offset by a constant distance --the orbit radius (Ror), as shown in Fig. 2. It is quite evident that the movement traces of all points in the circular arc B' C', the modified part of the orbiting scroll, are the circles with equal radius Ror as shown in Fig. 2. Since R=Ror +r, the envelope of those movement traces of all points on the circular arc B' C' is none other than the arc AB. This implies that the circular are B' C', the modified part of orbiting 480

4 scroll, and the circular are AB, the modified part of scroll, are in conjugate relation and can mesh with correctly. the fixed each other 2.Mathematical Analysis of The Conjug.ate Relation of The Modified Part The meshing engagement between the orbiting scroll and the fixed scroll in scroll compressors is a problem of contact transmission between scroll surfaces. By studying the plane meshing at the constant speed ratio based on the envelope theory for plane curves, we can obtain the conjugate profile of a given profile. This method can also be used in the ease where the speed ratio is varying. As shown in fig. 3, C <;L is a family of curves on a plane. L is such a curve, for its any point there is only one unique curve that be 1 on g s to the f ami 1 y of e u r v e s C.,.. and i s tangent i a l to L at this point. In addition, there are different points of tangency in curve L for different curves in the family. On the concept of envelope, the curve L is called the envelope of the family of curves C... The family of curves here includes any instantaneous position of the orbiting scroll profiles when it moves in the scroll compressor. If its envelope is just the profile of the inner surface of the fixed scroll AB, the meshing surfaces of orbiting scroll and the fixed scroll are in conjugate relation. As shown in fig. 4, the modified part of fixed scroll wrap is the circular are AB with F as its center and with the length of R as its radius. The trace of point E ' which is the center of circular are C' B', an arbitrary curve of the family of curves, forms a circle with the length of Ror as the radius when the orbiting scroll works. So we can get the equation of circular arc C' B' in rectangular coordinates system XOY: x=ror cos a o+r cos e y=ror sinao+r sine Obviously, the equation of the family of curves in the 481

5 rectangular coordinates system XOY is: c 0,: x=ror o cos a +r o cos e as follows: y=ror sin a +rosine If 1 is the envelope ofcu.,the equation of 1 can be written 1: x=x(e,a) y=y(e, a) Where a is a variable, e is varying with a., and it can be written as: e =f(a. ). Thus, the equation of the envelope 1 becomes: 1: x=x(f(a.), a.) y=y(f(a ), a.) Because 1 and C.,. are tangent to each other at corresponding points, the gradient of L and C... at these points must be equal. That is, k=ka., where k and ka. are gradient of Land Ca. on the points of tangency. We ean also obtain the following equation by derivating the equation of Land C"" separately, and using the chain rule. Ql X ;)Y -- = 0 de J a. Thus by solving the equations as f ollows: We can get: x=ror c 0 s a. + r. cos e j) y=ror s in a. + r s in e i~; CJY C) X C)Y ax -- - = 0 l:..,.: da. d e ae d a. a = e 482 '3\

6 Substituting a.= e into equations (}.i and (2), we obtain I "' (Ror+r) cos e y "' (Ror+r) sine Obviously, the above equations represent a circle whose center is point F, and radius is Ror+r.This circle is identical to the previous modified curve segment AB. That means the circular arc AB and C' B' are in conjugate relation. CONCLUSION From the graphic and mathematical analysis of the conjugate relation of the modified part of the fixed and orbiting scroll wrap, we can draw the conclusion that the modified part of the fixed and obiting scroll wrap produced by the new modification method presented at the l~st conference is in conjugate relation, and the new method is correct and reliable. ACKNOWLEDGEMENT The authors gratefuly acknowledges the support of K. C. WONG Education Foundation,HONG KONG. REFERENCES 1. Liu Zhenquan, Du Guirong, Yu Shicai, Wang Mingzhi, 'The Graphic Method of Modified Wrap of Scroll Compressor Proc. of the 1992 International Compressor Engineering Conference at Purdue pp James W. Bush, 'Derivation of A General Relation Conjugacy of Scroll Profiles" Proc. of the 1992 Compressor Engineering Conference at Purdue Governing The International pp ""' 3. Liu Zhenquan, Du Guirong, Yu Shicai, 'The Mathematical Derivation and Analysis of The Graphic Method For Wrap Modification of the Scroll Compressor' proc. of Xian Intern at ion a 1 Compressor Engine eying Conference I p p

7 Fig. 1 The Dluatrat1on of the graphical method for mod1flcatfon of the scroll wrap Flg.2 The envelope of moving traces of all points on the circular arc B'C' x X Ca. Fig.3 The envelope of all profiles of obiting scroll when it works Fig.4 The location of the points of arc 8' C' in XOY 484